From: scott@math.csuohio.edu (Brian M. Scott)
Newsgroups: sci.math
Subject: Re: Product of Normal Spaces
Date: Sat, 28 Feb 1998 02:37:32 GMT
On Fri, 27 Feb 1998 15:04:52 -0800, Wai-Shun Cheung
wrote:
>Could anybody give me an example that the product of two normal spaces
>is not normal?
Probably the simplest example is the Sorgenfrey line, S. The
underlying set is the real line, and the set of all intervals of the
form [x, y) is a base for the topology. It's easy to see that S is
Tikhonov. With a little more work one can show that S is Lindeloef
and therefore normal. But S x S isn't normal. To see this, let D =
{(x, -x) : x in R}, and note that D is a closed, discrete set in
S x S. Let F = {(x, -x) : x is rational}, and let K = D\F. Then F
and K are disjoint closed sets that can't be separated by disjoint
open sets. (Use the Baire Category Theorem in R to show that if V is
any open set containing K, there is an e > 0 and an interval (a, b) in
R such that for each irrational y in (a, b), [y, y+e) x [y, y+e) is
contained in V. If q in (a, b) is rational, this leaves no room for
an open nbhd of (q, -q) disjoint from V.)
Brian M. Scott
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Product of Normal Spaces
Date: 28 Feb 1998 20:27:19 GMT
In article <34F74694.3A0@math.uvic.ca>,
Wai-Shun Cheung wrote:
>Could anybody give me an example that the product of two normal spaces
>is not normal?
More than you wanted to know, taken from Math Reviews:
95b:54007 54A35 03E35 54D15 54D20 54G20
Szeptycki, Paul J.(3-TRNT); Weiss, William A. R.(3-TRNT)
Dowker spaces. (English) The work of Mary Ellen Rudin (Madison, WI,
1991), 119--129,
Ann. New York Acad. Sci., 705,
New York Acad. Sci., New York, 1993.
_________________________________________________________________
From the introduction: "A Dowker space is a $T\sb 1$, normal space
whose product with the closed unit interval is not normal. Whether
Dowker spaces exist was settled in 1970 by M. E. Rudin, who
constructed the only known example. Her space is large; it is a
$P$-space ($G\sb \delta$ are open) and it is of size and weight
$\aleph\sp \omega\sb \omega$. Instead of putting the subject to rest,
Rudin's construction gave rise to a flurry of activity in search of a
`small' Dowker space, that is, one whose important cardinal functions
(density, character, size) are small. Separable, first-countable
Dowker spaces of size continuum and $\omega\sb 1$ have been
constructed under a variety of extra set-theoretic assumptions
including the continuum hypothesis (CH), MA, the existence of a Suslin
tree, and variations of $\lozenge$. Rudin's article from the Handbook
of set theoretic topology is an excellent survey of the subject up to
1984. However, a `real' small Dowker space, that is, a small Dowker
space constructed with no further assumptions beyond ZFC, has eluded
us. In fact, there have been no significant advances on the two
central questions raised in Rudin's article: Are there `real' Dowker
spaces with small cardinal functions? And are there Dowker spaces with
strong global countable structures under any assumptions?
"We give a survey of what has been done in the last eight years,
revisit both Dowker's theorem and Rudin's space, and give a list of
open problems."
91f:54012 54D15 54B10
Hoshina, Takao(J-TSUKS)
Normality of product spaces. II. (English) Topics in general topology,
121--160,
North-Holland Math. Library, 41,
North-Holland, Amsterdam, 1989.
_________________________________________________________________
This is a survey, with proofs, of results concerning normality in
product spaces, with an emphasis on the following three conjectures of
K. Morita : (I) $X\times Y$ is normal for each normal space $Y$ if and
only if $X$ is discrete; (II) $X\times Y$ is normal for each normal
$P$-space $Y$ if and only if $X$ is metrizable; (III) $X\times Y$ is
normal for each countably paracompact space $Y$ if and only if $X$ is
metrizable and $\sigma$-locally compact.
The paper begins with a discussion of several basic, classical results
on normality in products, especially products with a compact factor.
The proof of conjecture (I) is given in Section 2; this consists of
the author's result that (I) holds if there is a $\kappa$-Dowker space
for each cardinal $\kappa$, together with M. E. Rudin's construction
of $\kappa$-Dowker spaces. Section 3 is devoted to the proof of K.
Chiba , T. Przymusinski and Rudin that conjecture (II) (and hence also
conjecture (III)) holds under $V=L$.
91f:54011 54D15 54B10
Atsuji, Masahiko(J-JOS)
Normality of product spaces. I. (English) Topics in general topology,
81--119,
North-Holland Math. Library, 41,
North-Holland, Amsterdam, 1989.
_________________________________________________________________
The author gives a survey, with proofs, of results on normality in
products, with an emphasis on products with a metric or generalized
metric factor. The first section deals with the metric factor case. It
contains, e.g., the result of Morita, and Rudin and Starbird, that if
$X$ is normal and countably paracompact and $Y$ is metrizable, then
$X\times Y$ is normal if and only if $X\times Y$ is countably
paracompact. Rudin and Starbird's theorem that if $Z$ is metrizable,
$X\times Z$ normal, and $f\:X\to Y$ a closed map, then $Y\times Z$ is
normal is also presented. The second section deals with generalized
metric factors. In particular, it contains results of the author that
some of the results of the first section remain valid when the metric
factor is replaced by a Lashnev (= closed image of a metric space)
factor.
The work by Rudin is
45 #2660 54D15 02K30 04A15
Rudin, Mary Ellen
A normal space $X$ for which $X\times I$ is not normal. (English)
Fund. Math. 73 (1971/72), no. 2, 179--186.
(also announced in Bull. Amer. Math. Soc. 77 (1971) 246, MR 42 #5217)